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Solving l1 regularized optimization problems is common in the fields of computational biology, signal processing and machine learning. Such l1 regularization is utilized to find sparse minimizers of convex functions. A well-known example is the LASSO problem, where the l1 norm regularizes a quadratic function. A multilevel framework is presented for solving such l1 regularized sparse optimization problems efficiently. We take advantage of the expected sparseness of the solution, and create a<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1607.00315v1">arXiv:1607.00315v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/32v7bkfjy5hbza5ldzgkc4gq5i">fatcat:32v7bkfjy5hbza5ldzgkc4gq5i</a> </span>
more »... rarchy of problems of similar type, which is traversed in order to accelerate the optimization process. This framework is applied for solving two problems: (1) the sparse inverse covariance estimation problem, and (2) l1-regularized logistic regression. In the first problem, the inverse of an unknown covariance matrix of a multivariate normal distribution is estimated, under the assumption that it is sparse. To this end, an l1 regularized log-determinant optimization problem needs to be solved. This task is challenging especially for large-scale datasets, due to time and memory limitations. In the second problem, the l1-regularization is added to the logistic regression classification objective to reduce overfitting to the data and obtain a sparse model. Numerical experiments demonstrate the efficiency of the multilevel framework in accelerating existing iterative solvers for both of these problems.
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